Students Can Learn to Be Better
Problem Solvers
Arthur Whimbey
Students at every level of education from ele
mentary school to medical school are learning
problem-solving skills. One of the secrets?
"Be careful"
Why are some high school and college students
better than others in comprehending and solving
problems like this:
If deleting the letters r, i, and e from the word sur
mise leaves a meaningful three-letter word, circle the first
s in this word surmise. Otherwise circle the u in the word
surmise where it appears for the third time in the exercise.
It is important to know why some students are
better than others at solving this type of problem, be
cause it taps the same analytical skills a person needs
to master content and solve problems in mathematics,
chemistry, or other sciences, and to read tax forms,
technical reports, and so much else that is part of a
literate and technological culture.
As early as 1950 Benjamin Bloom and Lois
560
EDUCATIONAL LEADERSHIP
Broder (now Greenfield) saw that a major barrier to
understanding and teaching problem solving is that
thinking is generally done inside a person's head
where it is hidden from view. This makes it difficult
for a teacher to teach and for a learner to learn. A
beginner cannot observe how an expert thinks and
solves problems in the way the beginner might observe
a golf pro taking a stance, gripping the club, then
executing the swing. And the expert cannot observe
the beginner practicing, pointing out flaws and show
ing him/her how to improve.
Thinking Aloud
A way around this barrier is to have people
think aloud while they solve problems, so their
methods can be observed. When Bloom and Broder
had scholastically successful and unsuccessful stu
dents think aloud as they solved problems, a striking
difference was observed. Unsuccessful students were
mentally careless and superficial in solving problems.
They often rushed through a problem and selected a
wrong answer because they failed to comprehend
what was required. When urged to reread the prob-
lem carefully, they often understood it better and
proceeded correctly. Even then, however, they were
not thorough but settled for a rough interpretation.
Also, they tended to be passive in their think
ing. They spent little time considering a question but
chose an answer on the basis of a few clues, such as
a feeling, an impression, or a guess.
By contrast, successful students made an active
attack on problems. When a question was initially
unclear, they often employed a lengthy sequential
analysis in arriving at an answer. They began with
what they understood, drew on other information
in their possession, and went through a chain of
steps that finally brought them to a solution.
The approach of the unsuccessful student was
like that of a novice biology student who hadn't
learned to look carefully through a microscope, dis
tinguish the various forms, and put together an
accurate picture of the material on the slide.
In order to show students the activities of good
problem solvers, Jack Lochhead, director of the
Cognitive Development Project at the University of
Massachusetts, and I have developed a workbook
that includes thinking aloud responses of successful
students and professionals. Here is the response of a
successful medical student to the problem shown
earlier.
Problem Solver's Response
Problem Solver began reading aloud.
If deleting the letters r, i, e, from the word sur
mise . . .
Problem Solver repeated the letters aloud.
r, i, e.
Problem Solver crossed out the letters r, i, e w ith her pen
cil. She then read the remaining letters aloud and pro
nounced the word they formed.
s, u, m, s . . . sums.
Problem Solver resumed reading.
leaves a meaningful three-letter word, circle . . .
Problem Solver stopped reading and thought aloud.
I'm getting to the section where I have to do some
thing if the first part is true. I'm a little confused
though, maybe I should start reading the sentence
from the beginning. No, I'll read the rest of it.
circle the first s in this word surmise.
I am confused, let me start again.
If deleting the letters r, i, and e from the word sur
mise leaves a meaningful three-letter word . . .
No, sums is a four-letter word.
Circle the first s in this word surmise.
So I won't do this.
Otherwise circle the u in the word surmise where it
appears for the third time in the exercise.
So this is the direction I'm supposed to follow. Let's
see, the word surmise appears once . . . twice in the
first sentence. Then in the second sentence it appears
three and that's it. It appears for the third time in
the second sentence. So that's the one I should circle.
Now I'll reread the last sentence to make sure I fol
lowed the directions correctly.
Otherwise circle the u in the word surmise where it
appears for the third time in the exercise.
I'll scan the sentence again to be sure that the word
surmise doesn't occur anywhere except the three
places I found it. Yes, / circled the right one. 1
This response illustrates two major characteris
tics that distinguish successful from unsuccessful
students: the step-by-step approach; and carefulness
the concern and quick retracking when ideas be
come confusing, the rechecking, reviewing, and re
reading to be sure that errors haven't crept in, that
nothing is overlooked.
Teaching Reasoning Through Discussion
In teaching a skill such as golf, the instructor
demonstrates the correct performance and then en
gages the student in extended, guided practice. To
provide a comparable experience in teaching analyti
cal reasoning, educators have shifted away from the
frontal, lecture-oriented class to an active, discussionoriented format. They recognize that while lectures
have their place, if teachers lecture for a major
portion of the class they are not monitoring and
guiding the development of analytical skills.
In the original Bloom and Broder (1950) study,
unsuccessful students worked in small groups with a
tutor, taking turns solving problems aloud, reading
the solutions of successful students, and attempting
to correct their weaknesses. They found that by
thinking aloud they not only communicated their
thoughts to others, but they became more aware of
gaps and inadequacies in the way they approached
problems. Bloom and Broder observed systematic
improvement over the training sessions; students
read problems with greater care, and reasoned more
actively and accurately. The group also showed a
statistically significant gain on the University of
Chicago's comprehensive examination, which consists
of problems in various academic areas.
The procedure of having students work in pairs
or small groups and explain their thinking is now
being successfully employed in schools ranging from
the elementary level through college. Some classes
use the workbook which Lochhead and I developed.
In physics and engineering classes at the University
of Massachusetts, students spend several hours at
the beginning of the semester working in pairs and
1 Reprinted from A. Whimbey and J. Lochhead, Problem
Solving and Comprehension: A Short Course in Analytical
Reasoning (Philadelphia, Pa.: Franklin Institute Press, 1979).
APRIL 1980
561
thinking aloud as they solve problems like the one
shown earlier. Then they switch to physics or engi
neering problems but continue to work in pairs aloud.
In contrast, various high schools and colleges will
have weaker students spend the entire semestir solv
ing such problems. They focus entirely on analytical
skills without involving specific content from physics,
mathematics, or other subjects.
Project SOAR (Stress On Analytical Reasoning),
a five-week pre-first year program at Xavier Univer
sity in New Orleans, included Piagetian-based science
laboratories in the morning, and vocabulary expan
sion, problem solving, and reading exercises using the
thinking aloud format in the afternoon. These gains
were reported by SOAR's director:
The 34 students who initially scored below grade 12
on the comprehension section of the Nelson-Denny Read
ing Exam showed an average improvement of 1.4 years.
The 43 students who scored below grade 12 on the
vocabulary section of the Nelson-Denny gained an aver
age of 1.8 years.
The 21 students who scored less than 70 on the PSAT
(equivalent to 700 on the SAT) gained an average of 11.4
points, and the entire group (113 students) gained 7.3
points. These were equivalent to gains of 114 and 73,
respectively, on the SAT (Carmichael, 1979, p. 2).
In 1960 a front-page article in The New York
Times reported an average IQ increase of 10.5 points
for the 280 first year students at Whittier College
who took a two semester course designed by English
professor Albert Upton. The pedagogy of Upton's
program was very similar to that of Bloom and
Broder. Classes met four times a week; once for
lecture and three times in small groups with tutors
for problem solving with verbal comprehension and
reasoning problems.
Upton's approach has been incorporated into a
remedial language arts program called THINK, span
ning reading levels one-12, and a parallel math pro
gram called Intuitive Mathematics.2 The problem
solving/discussion format used by both programs
is proving very effective. In studies by June Gabler
(1977), superintendent of the Woodhaven school
district in Michigan, THINK produced larger gains
on the Nelson-Denny than those produced by stand2 Both programs are available from ISI Think, Inc., 300
Broad St., Stamford, CT 06901.
Problem Solving in Mathematics: National Assessment Results"
An important goal of the mathematics curriculum
is to teach students to apply the mathematics they are
learning to new or unfamiliar situations. Recently
released results from the second mathematics assess
ment of the National Assessment of Educational Prog
ress (NAEP) indicate that although students are learn
ing many basic algorithmic or computational skills,
they have difficulty applying these skills to solve even
simple nonroutine problems.
The NAEP mathematics assessment results are
based on the performance of a representative sample
of over 70,000 9-, 13-, and 17-year olds who took a
carefully developed set of about 500 exercises that
assessed important objectives of the mathematics cur
riculum. The results of this assessment represent the
best available measure of American students' mathe
matical achievement.
Students were generally successful in solving sim
ple textbook problems that could be solved by apply
ing a single operation to the numbers given in the
problem. However, any exercise that required students
to do more than decide whether to add, subtract, mul
tiply, or divide caused considerable difficulty.
At all three age levels, students would frequently
attempt to apply a single mathematical operation to
whatever numbers were given in a problem rather
than analyzing the problem to decide how to solve it.
For example, only about 10 percent of the 9-year-olds
562
EDUCATIONAL LEADERSHIP
and 30 percent of the 13-year-olds correctly solved
the following problem:
Mr. Jones put a wire fence all the way around his
rectangular garden. The garden is 10 feet long and 6 feet
wide. How many feet of fencing did he use?
Almost 60 percent of the 9-year-olds and 40 percent
of the 13-year-olds simply added the 6 and the 10.
An important aspect of problem solving is iden
tifying which facts are relevant to a given problem.
The following item, which includes extraneous data,
illustrates the difficulty students had in analyzing a
problem.
One rabbit eats 2 pounds of food each week. There
are 52 weeks in a year. How much food will 5 rabbits
eat in a week.
Almost a fourth of the 13-year-olds1 attempted to in
corporate all three numbers given in the problem into
their calculation.
When students could identify the appropriate
operation, they frequently had difficulty relating the
result of their calculation to the given problem. The
following exercise involves a simple application of
* This material is based upon work supported by the National
Science Foundation under Grant No. SED-7920086. Any
opinions, findings, and conclusions or recommendations ex
pressed in this publication are those of the authors and do not
necessarily reflect the views of the National Science Foundation.
ard reading instruction for below average, average,
and above average students in both middle and high
schools. Also, Intuitive Mathematics resulted in
average gains on the Metropolitan Achievement Test
which were three times those from standard instruc
tion for high school students with histories of diffi
culties in mathematics (Gabler, 1977).
Taking a somewhat different tack, Jill Larkin
(1977) compared graduate students with outstanding
physicists at Berkeley on the way they solved certain
physics problems. By having them think aloud she
found that the graduate students tended to be "form
ula bound" while the physicists used mental pictures
and other analytical devices to understand situations
before starting their computations. Booklets were
then developed which successfully taught the analyti
cal strategies to introductory physics classes. In this
case the training did not involve small discussion
groups; students worked alone, but the booklets did
require active responses from the students as they
practiced the mental activities reported by the out
standing physicists. Research indicates that small
group discussion is useful at all ability levels but is
long division. But the answer to the problem is the
remainder to the division calculation, not the quotient.
A man has 1310 baseballs to pack in boxes which
hold 24 baseballs each. How many baseballs will be left
over after the man has filled as many boxes as he can?
Fewer than 30 percent of the 13-year-olds correctly
answered this problem. Over 20 percent gave the quo
tient as their answer.
Students' mechanical application of computa
tional algorithms often resulted in unreasonable an
swers that they should have recognized if they had
thought about the problem. For example, consider the
multiple choice estimation exercise in Figure 1. Many
of the students at each age level simply added the
numbers in the numerator or denominator. The magni
tude of the result is completely unreasonable if one
understands what it means to add two fractions less
Figure 1 Sample Estimation Problem
ESTIMATE the answer to
Response
0
0
0
0
0
1
2
19
21
I don'l know
— + J13
+
8
-
Percent Responding
Age 17
Age 13
7
24
28
26
14
8
37
21
15
16
most crucial for the students at the lower levels.
The emphasis on thinking skills is reaching the
highest educational settings, as illustrated in these
excerpts from a description of a program at McMaster
University School of Medicine, written by the direc
tor, Howard Barrows (1979).
The problem solving skills of physicians are central
to their ability to apply health care effectively and effi
ciently. McMaster University School of Medicine has
chosen problem based, self directed learning in small
groups as the principal format for learning undergraduate
medicine. One of the objectives of problem based learning
is to help students develop appropriate cognitive skills
in medical problem solving. Therefore, a group of us in
the School of Medicine completed a five-year study of the
problem solving process of the physician.
The next stage of our work was to design appropriate
learning units to develop this problem solving process in
postgraduate and continuing medical educational settings.
At the present time we are formulating an overall
sequence for the development of the problem solving
process of the medical student from small group learning
around the simulated patient, through individual tasks
with simulated patients, ending with structured clinical
situations with real patients under pressure to finely tune
the problem solving process.
than one. However, rather than estimating the sum,
many students appear to have attempted to find some
calculation involving the numbers given in the exercise
with no concern for the reasonableness of the result.
Performance was over 15 percentage points better on
a direct computation exercise in which students were
asked to calculate the sum of the two fractions.
The results for the exercises described here as
well as many other exercises on the assessment
strongly suggest that students have become accus
tomed to mechanically applying mathematical algo
rithms to problems without analyzing the problems or
thinking about the reasonableness of their answers. In
recent years, a great deal of attention has focused
on the teaching of basic skills, which are frequently
equated with routine computational skills. Mathe
matics skills, however, have little value if they cannot
be applied to new or unfamiliar situations.
The ability to analyze a problem situation is as
important and as basic a skill as the ability to compute
the answer. The assessment results suggest that prob
lem solving is the basic skill most in need of attention
in the mathematics curriculum.
Thomas P. Carpenter, Mary Kay Corbitt, Henry
Kepner, Mary Montgomery Lindquist, Robert E.
Keys.
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563
Problem Solving In Every Class
Many schools are now including a course in
problem solving as a part of the curriculum. This is a
good start, but one course in problem solving cannot
be expected to make students expert thinkers, any
more than a single course in golf or drawing produces
experts. Attitudes and activities of analytical reason
ing should be reinforced throughout the curriculum.
One example of this reinforcement is the ex
perimental high school program pioneered by Carl
Frankenstein of Hebrew University's department of
special education (Peleg and Adler, 1977). The Frank
enstein program, now incorporated in 42 Israeli
schools, instructs teachers in how to engage students
in Socratic dialogue. Whether in mathematics, science,
or Bible study, whenever students answer questions
incorrectly, they are asked to explain their reasoning.
This may mean going back to the text to read and
interpret pertinent sections or detailing the operations
and logic used in reaching an answer. With teachers
in all classes focusing on accuracy and social workers
giving parents information on providing a supportive
home environment, dramatic improvements in cogni
tive skills have been obtained by Frankenstein and
his associates.
Bloomfield College of New Jersey is also stres
sing mental processes in its first year core program.
In describing Bloomfield's program, Hechinger (1979)
wrote in C hange m agazine that the "process is the
same whether a student is reading a book or doing
chemistry the stress is on systematic analysis and
logical examination of problems . . . whatever the
subject, there is always a heavy concentration on the
analysis of a relatively few pages."
Venezuela has taken such a great interest in the
improvement of thinking skills that there is now a
Minister of State for the Development of Human
Intelligence. More modest efforts can also be effec
tive. A political science class at Bowling Green State
University switched from three lectures per week to
two lectures and two labs. During the lab hours stu
dents read selections from the text and answer ques
tions in the accompanying workbook. They then
compare answers with a partner, and where there
is disagreement, defend the answer by citing sections
from the text and explaining their reasoning. In this
way students master the course content while
strengthening analytical skills.
Be Careful
of educational research seems to have caught nature
at this game. It suggests that one of the most im
portant lessons teachers can impart to students is
simply: "Be extremely thorough and careful in your
thinking." For years we urged students not to count
on their fingers, not to move their lips when they
read, and to try to read faster. Research has now
shown that there are engineers who count on their
fingers, lawyers who move their lips while reading,
and literary figures like William Buckley who admit
they read "painfully slowly." What we have not done
is stressed emphatically to students that the core of
academic success is systematic, accurate thought.
Newton modestly observed:
If I have succeeded in my inquiries more than others,
I owe it less to any superior strength of mind, than to a
habit of patient thinking.
We are now beginning an educational experiment
in teaching this "habit of patient thinking." As teach
ers from the first grade through college begin to echo
insistently "Be accurate" and "Explain how you
obtained that answer," it will be exciting to see how
high we can raise the national levels of reading,
mathematics, and problem-solving skills. K
References
Barrows, H. "Problem Based Learning." Problem Solving I
(November 1979).
Bloom, Benjamin S ., and Broder, Lois. Problem-solving
Processes of College Students. Chicago: University of Chicago
Press, 1950.
Carmichael, J. W. 'Teaching Critical Reading and Analy
tical Reasoning In Project SOAR," 1979. (Mimeographed.)
Gabler, J. "Summary of Test Findings on Think Pilot,"
1977. (Mimeographed.)
Hechinger, F. "Basic Skills: Closing The Gap." Change
(October 1979): 29-33.
Larkin, J. H. "Processing Information for Effective Prob
lem-Solving." Berkeley: Department of Physics and Group in
Science and Mathematics Education, University of California,
1977. (Mimeographed.)
Peleg, R., and Adler, C. "Compensatory Education in
Israel: Conceptions, Attitudes and Trends." American Psy
chologist 32 (1977): 94S-S8.
Whimbey, A., and Lochhead, J. Problem Solving and Com
prehension : A Short Course in Analytical Reasoning. Philadel
phia: Franklin Institute Press, 1978.
Arthur Whimbey is Professor
of Education at Clark Col
lege, Atlanta, Georgia,
Nature enjoys playing an occasional trick on
humans, and one of her favorite tricks is to make an
important discovery or principle too simple to be
recognized as important. However, a growing body
APRIL 1960
565
Copyright © 1980 by the Association for Supervision and Curriculum
Development. All rights reserved.